ON (α – β) - CONTRACTIVE MAPPING OF PARTIAL b-METRIC SPACES AND FIXED POINT THEOREM

Abstract

Abstract

we prove in this article a unique common fixed point theorem for α – β contractive condition for rational contraction and also we give example that explain the main result.

Keywords:  Partial, b-metric space, weakly , compatible mapping, rational contraction.  metric space.

1.Preliminaries

In the year 1992,a partial metric space is a generalization of the notion of the metric space defined in 1906,by (Maurice.frechet) such that the distance of a point from itself isn't necessarily 0.well known banach fixed point theorem also known as banach contaction principle,was.a foundation for a development of metric fixed point theory and found applications in different areas. There was much popularization of that result in tha last seventy years.At 1989 [1] submitted the notion of Quasi-Metric Space as a special general concept of Metric Spaces. At1993,[2][3] put many theorems due to the b–metric spaces.In (1994)[4] assumptive the notion of partial metric space in which the Self-distance of every point of space not equal 0.At(1996)o'neill generalized the concept of (P.M.SP) by introduced negative distances.At(2013) [5] popularized both the notion of (b-M.SP) and (P.M.SP) by presenting the partial b-metric spaces (P.b-M.SP)as example alot of researchers at present day have studying this presupposition and its generalization in different types of (M.SP).

some authors close to our interest were studying some fixed point theorems in the so called b-metric space. After then, some authors started to prove ((α – ψ)) versions of certain fixed point theorems in different type metric spaces [6,7,8]. Mustafa in [9], gave a generalization of Banach's contraction principles in a complete ordered partial b-metric space by introducing a generalized (α-ψ)weakly contractive mapping.